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Notice that we can get the equation of line \(k\) which is perpendicular to line \(y=2x\) if we know ANY point that line \(k\) passes through. So, to get the equation of line \(k\) we need the values of \(a\) and \(b\).

(1) \(a = -b\). Not sufficient.

(2) \(a - b = 1\). Not sufficient.

(1)+(2) \(a = -b\) and \(a - b = 1\), we have two distinct linear equations with two unknowns so we can solve for \(a\) and \(b\). Sufficient.

can't A suffice to find solution? y=2x is given as going from quad 3 through origin to quad 1 with slope 2.

slope of the perped line is -1/2, relatioship between points is given as whenever a is positive the b bears the same value but with opposite sign (10,-10 or -10;10) - hence the second line has to go through the origin from quad 2 to quad 4?

can't A suffice to find solution? y=2x is given as going from quad 3 through origin to quad 1 with slope 2.

slope of the perped line is -1/2, relatioship between points is given as whenever a is positive the b bears the same value but with opposite sign (10,-10 or -10;10) - hence the second line has to go through the origin from quad 2 to quad 4?

thanks

Why should line k go through the origin?
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I think this is a high-quality question and the explanation isn't clear enough, please elaborate. Bunuel,

If the intention is merely to find values of a and b then what is the significance of "Perpendicularity" here? Please explain

The question asks to find the equation of k that is perpendicular to line y=2x and passes through point (a,b). We need to find the values of a and b to answer the question.
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S1) take different values for a,b that meet the given statement (a=-b) and form lines equations of lines with slope -1/2. You can see that for different values of (a,b), you get different equations.

S2) same as statement 1

Both) Both equations give a point a=1/2 and b=-1/2. Which hivea only one equation 2x+4y=-1

A little clarification equation of line perpendicular to y-2x=0 will be -2y-x+c=0 ?? In order to find equation of a line perpendicular to ax+by+c=0.....interchange coefficients of x and y and change the sign in between....so equation of line perpendicular to ax+by+c=0 will be bx-ay+k=0. Is this approach correct?

If the question were to be a slightly different one, would the below Statement be sufficient?

Question stem: Find equation of the line K, perpendicular to a line y= -1/2x and passes through point (a,b).

statement 1) 2a=b

I understood the point that given a slope + either the y-intercept or a specific point, we can plot the specific line. Where I am getting confused is whether we can still identify the line by having the slope and a generic equation such as 2a=b.

When I tried plotting several lines with the slope = 2 and with different y-intercepts, I could not come up with any point that satisfy 2a=b and any other lines except y=2x+0.

Would you please help me clear this point? Thanks!

I do not understand why we cannot say that stat I implies a=b=0, hence being sufficient while statement II implies basically all consecutive integers, not being sufficient. Can someone please explain?

I do not understand why we cannot say that stat I implies a=b=0, hence being sufficient while statement II implies basically all consecutive integers, not being sufficient. Can someone please explain?

a = -b does not necessarily means that a = b= 0. Why not a= 1 and b = -1?
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